论文标题
三阶la脚的单肌
Monodromy of a generalized Lame equation of third order
论文作者
论文摘要
我们研究以下三阶线性差分方程的单片\ [y''（z） - （α\ wp（z;τ）+b）y'（z）+β\ wp'（z;τ）y（z）= 0，\]，其中$ b \ in $ b \ in \ n \ in \ mathbb {c} agars a parameter a parameter a parame; $ \ wp $ - 功能$ 1 $和$τ$，以及$α，β$是常量，使得奇异性$ 0 $的本地指数是三个独特的整数，可以始终将其写入$ -n-l，1-l，1-l，1-l，n+2l+2 $，经过双重变换后，$ n，l，l \ n $ n，l \ in \ in \ n Mathbb bb bb bb bb bb bb bb bb bb bb bb n n} $ n} $。该颂歌可以看作是众所周知的lamé方程的三阶版本$ y''（z） - （m（m+1）\ wp（z;τ）+b）y（z）= 0 $。我们说，如果单一组与单一组的亚组偶联，则单一构度是统一的。我们证明\ begin {inatizize} \ item [（i）]如果$ n，l $都是奇怪的，那么单型不可统一； \ item [（ii）]如果$ n $是奇数，而$ l $甚至是$ b $，则$ b $的有限值是单型元素是klein四组，因此是统一的； \ item [（iii）]如果$ n $甚至是$ n $，那么是否存在$ b $，以使单一元素统一取决于选择$τ$的选择。 \ end {inatize}研究二阶拉梅方程的方法在这里无法使用，我们需要开发不同的方法来分别处理这些不同的情况。这些结果在另一项作品（Chen-lin，J。Dill。Geom。
We study the monodromy of the following third order linear differential equation \[y'''(z)-(α\wp(z;τ)+B)y'(z)+β\wp'(z;τ)y(z)=0, \] where $B\in\mathbb{C}$ is a parameter, $\wp(z;τ)$ is the Weierstrass $\wp$-function with periods $1$ and $τ$, and $α,β$ are constants such that the local exponents at the singularity $0$ are three distinct integers, which can always be written as $-n-l, 1-l, n+2l+2$ after a dual transformation, where $n,l\in\mathbb{N}$. This ODE can be seen as the third order version of the well-known Lamé equation $y''(z)-(m(m+1)\wp(z;τ)+B)y(z)=0$. We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that \begin{itemize} \item[(i)] if $n, l$ are both odd, then the monodromy can not be unitary; \item[(ii)] if $n$ is odd and $l$ is even, then there exist finite values of $B$ such that the monodromy is the Klein four-group and hence unitary; \item[(iii)] if $n$ is even, then whether there exists $B$ such that the monodromy is unitary depends on the choice of the period $τ$. \end{itemize} The methods of studying the second order Lamé equation can not work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable $SU(3)$ Toda system in another work (Chen-Lin, J. Differ. Geom. to appear).
